Question

draw the image of △abc under a dilation whose center is p and scale factor is 3.

Explanation:

Step1: Recall Dilation Rules

Dilation with center \( P \) and scale factor \( k \) means for each point \( X \) in \( \triangle ABC \), we find \( X' \) such that \( \overrightarrow{PX'} = k \cdot \overrightarrow{PX} \) (or \( PX' = k \cdot PX \) and \( X' \) lies on the line \( PX \)). Here, \( k = 3 \).

Step2: Locate Center \( P \)

Identify the position of point \( P \) on the grid. Then, for each vertex \( A \), \( B \), \( C \) of \( \triangle ABC \):

• For Point \( A \):
• Draw the line from \( P \) through \( A \).
• Measure the distance from \( P \) to \( A \) (let's call it \( d_{PA} \)). Multiply this distance by \( 3 \) to get \( d_{PA'} = 3 \cdot d_{PA} \). Mark \( A' \) on the line \( PA \) such that \( PA' = 3 \cdot PA \).

• For Point \( B \):
• Draw the line from \( P \) through \( B \).
• Measure the distance from \( P \) to \( B \) ( \( d_{PB} \) ). Multiply by \( 3 \) to get \( d_{PB'} = 3 \cdot d_{PB} \). Mark \( B' \) on line \( PB \) with \( PB' = 3 \cdot PB \).

• For Point \( C \):
• Draw the line from \( P \) through \( C \).
• Measure the distance from \( P \) to \( C \) ( \( d_{PC} \) ). Multiply by \( 3 \) to get \( d_{PC'} = 3 \cdot d_{PC} \). Mark \( C' \) on line \( PC \) with \( PC' = 3 \cdot PC \).

Step3: Connect \( A' \), \( B' \), \( C' \)

Once \( A' \), \( B' \), and \( C' \) are marked, connect them to form \( \triangle A'B'C' \), which is the image of \( \triangle ABC \) under dilation with center \( P \) and scale factor \( 3 \).

(Note: Since this is a drawing task, the key is to use the dilation properties: collinearity with the center and distance scaled by 3. The lengths of \( \triangle ABC \) ( \( AC = 3 \), \( CB = 4 \), \( AB = 5 \)) will scale to \( A'C' = 9 \), \( C'B' = 12 \), \( A'B' = 15 \) as the scale factor is 3, maintaining the right - triangle shape, and the position relative to \( P \) is scaled by 3.)

Answer:

To draw the image:

1. Draw lines from \( P \) through \( A \), \( B \), and \( C \).
2. On each line, mark points \( A' \), \( B' \), \( C' \) such that the distance from \( P \) to each new point is 3 times the distance from \( P \) to the original point ( \( PA' = 3PA \), \( PB' = 3PB \), \( PC' = 3PC \) ).
3. Connect \( A' \), \( B' \), and \( C' \) to form the dilated triangle \( \triangle A'B'C' \). The side lengths of \( \triangle A'B'C' \) will be \( A'C' = 9 \), \( C'B' = 12 \), and \( A'B' = 15 \) (since \( 3\times3 = 9 \), \( 3\times4 = 12 \), \( 3\times5 = 15 \)), and it will be similar to \( \triangle ABC \) with the same shape, centered at \( P \) with scale factor 3.